Question:
In the buglary system, who is more reliable wen a buglary happened?
Mary or John?
Proof it by calculation using bayesian Network Theory.

References of values:

P(B)	0.001
P(¬B)	0.999
P(E)	0.002
P(¬E)	0.998
P(J|A)	0.90
P(¬J|A)	0.10
P(J|¬A)	0.05
P(¬J|¬A)	0.95
P(M|A)	0.70
P(¬M|A)	0.30
P(M|¬A)	0.01
P(¬M|¬A)	0.99
P(A|B^E)	0.95
P(A|B^¬E)	0.94
P(A|¬B^E)	0.29
P(A|¬B^¬E)	0.001
P(¬A|B^E)	0.05
P(¬A|¬B^E)	0.71
P(¬A|B^¬E)	0.06
P(¬A|¬B^¬E)	0.999

By using clausal inference in bayesian network:
(from lecture notes)

John calls when there's buglary happened.

P(J | B)
= P(J | A Λ B).P(A | B) + P(J | ¬A Λ B). P(¬A | B)
= P(J | A).P(A | B) + P(J | ¬A).P(¬A | B)
= P(J | A).P(A | B) + P(J | ¬A).(1 – P(¬A | B))
Alarm activated when there's buglary
P(A | B)
= P(A | B Λ E). P(E | B) + P(A | B Λ ¬E).P(¬E | B)
= P(A | B Λ E).P(E) + P(A | B Λ ¬E).P(¬E)
= 0.95 x 0.002 + 0.94 x 0.998 = 0.94002

P(J | B) = 0.90 x 0.94002 + 0.05 x 0.05998
= 0.849017

For Marry:
Marry calls when there's buglary happened:
P(M | B)
= P(M | A Λ B).P(A | B) + P(M | ¬A Λ B). P(¬A | B)
= P(M | A).P(A | B) + P(M | ¬A).P(¬A | B)
= P(M | A).P(A | B) + P(M | ¬A).(1 – P(¬A | B))

P(A | B)
= P(A | B Λ E). P(E | B) + P(A | B Λ ¬E).P(¬E | B)
= P(A | B Λ E).P(E) + P(A | B Λ ¬E).P(¬E)

0.95 x 0.002 + 0.94 x 0.998

0.94002
P(M | B)
= 0.7 X 0.94002 + 0.01 X 0.05998
=0.6586138

Comparing the two values of probability, i get that the probability of John calls when there's buglary happened is higher (0.849017), compares to Marry calls when there's buglary happening (0.6586138).
Therefore, John is more reliable when a buglary happened.

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